7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Revisiting the magnetic field and flow models in the planetary magnetosheath

Y. Narita1 and D. Schmid2
1Institute of Theoretical Physics, TU Braunschweig, 2 Space Research Institute, Austrian Academyof Sciencese-mail (speaker): y.narita@tu-braunschweig.de
The steady-state magnetosheath model has various
applications in studying the plasma and magnetic field
profile around the planetary magnetospheres. Such amodel can be used to evaluate the growth rate of plasma
instabilities (mirror mode, in particular) by tracking theplasma parcel along the streamline, to navigate the data
analysis, and even to estimate the upstream solar windcondition by inverting the model. The steady-statemagnetosheath model is analytically modeled byconstructing the Laplace equation for a scalar potentialthat cancels the solar wind field (such as the magnetic
field or the flow velocity field) in the planetary
magnetosphere [1]. The Laplace equation can be solved
in different ways, which lead to different modelingapproaches.
One may of course solve the Laplace equation
numerically with the current computing resource but theboundary values of the scalar potential surrounding themagnetosheath must be set by hand, which turns out tobe a difficult task because the magnetosheath isphysically bounded in the radial direction (by the bow
shock and the magnetopause) but not along the flow inthe magnetosheath.
Our approach is based on the fact that the Laplace
equation on the magnetosheath can exactly be solved toobtain an analytic expression of the scalar potential in themagnetosheath under a special condition, that is for theparabolic shape of the magnetosheath bounded by theparabolic bow shock and magnetopause [1]. The solution
is found by expanding the potential with the Bessel
functions and truncating at the second order due to theboundary condition. The exact solution can then be used
to model the magnetic field and flow velocity field for a
non-parabolic (i.e., more realistic) magnetosheath.
One possibility is to iterate in the stream-wise sense(along the flow) by tracking along the flow direction stepby step [2,3], which is computationally expensive due tothe need for many iterations and also for solving theshock condition ad hoc to obtain the initial condition
properly.
The other possibility is to implement a mapping of themagnetosheath coordinates by retaining the grids
orthogonal or nearly-orthogonal upon the coordinatetransformation. Conformal mapping (angle-preservingmapping) on the complex plane is the ideal method tothis goal, [4] but finding the proper conformal mapping
turns out to be a mathematical challenge due to the nonbounded nature of magnetosheath along the flow. We

propose an orthogonal grid system at and around themagnetopause and extend the grids to the bow shock(magnetopause-normal mapping) [5], which isadvantageous in the following aspects: (1) the quality of
scalar-potential mapping is high with the nearlyorthogonal nature of the mapping (Fig. 1), (2) thecomputation is numerically by far inexpensive comparedto the iteration method, (3) the mapping is applicable to auser-specified shape of bow shock and magnetopause,and (4) the algorithm is easy to implement.

Figure 1. Flow velocity potential, stream function,
magnetic scalar potential, and magnetic field directionsin the empirical Earth magnetosheath obtained bymagetopause-normall mapping [5].

Reference[1] Kobel, E., and Flückiger, E. O., J. Geophys. Res., 99,23617, 1994
[2] Soucek, J., and Escoubet, C. P., Ann. Geophys., 30,973, 2012[3] Schmid et al., A&A, 668, A113, 2022
[4] Narita et al., Ann. Geophys., 41, 87, 2023
[5] Narita et al., manuscript submitted to Ann. Geophys.